Extended Thesis Abstracts

Analyzing and Addressing Common

Mathematical Errors in Secondary Education

Megan Elbrink

Megan Elbrink graduated Magna Cum Laude fromBall State in December of 2007 with a major in Mathe-matics Education. She is currently substitute teachingand pursuing job opportunities within the EvansvilleVanderburgh School Corporation. Dr. Sheryl Stumpwas the advisor for this honors thesis.

After much research and discussion with mathematics educators, it is apparentthat secondary education students make many common errors in mathemat-ics. As a result, it is important to analyze what causes and how to addressthese mathematical errors so that teachers can correct and prevent these errorswithin the classroom. For my honors thesis, I briefly researched and analyzedthe causes of mathematical errors in secondary education and ways to addressstudents’ errors. Then, through research, classroom observations, and discus-sions with secondary education teachers, I compiled a list of common studenterrors in mathematics and created activity sheets that address each error in-dividually. To coincide with my research findings, the activity sheets confronteach error conceptually and incorporate various mathematical representations.

Categorizing Students’ Mathematical Errors

Based on my research, I was able to classify students’ mathematical errors intothree main categories: calculation errors, procedural errors, and symbolic er-rors. Calculation errors can be generalized as mistakes in addition, subtraction,multiplication, and division of numbers. Carelessness and lack of attention canresult in calculation errors. Procedural errors occur when a student computes

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or applies a procedure incorrectly. These types of errors suggest that studentsdo not understand the concept related to the procedure. As a result, studentsdo not have an understanding of why or how a procedure works; therefore, stu-dents do not recognize the importance applying and computing the procedurecorrectly. Lastly, symbolic errors occur when students falsely relate mathe-matical problems that use similar symbols. Students try to create meaning inthe patterns of mathematical symbols and signs that they see in front of themrather than trying to understand what they are actually doing. This searchfor patterns in the symbols leads to misinterpretations, which in turn result inmathematical errors. Based on this research, I was able to classify each errorthat I compiled into one of these three categories.

Addressing Common Mathematical Errors

Once I had identified three categories in which I could classify students’ math-ematical errors, I then researched how to address each type of error. First, itwas necessary to address calculation errors. Since these errors are often theresult of carelessness or short attention span, they are the easiest type of errorsto address. A possible solution is the incorporation of an error checklist intothe regular classroom routines and procedures. This allows students to assessthemselves and identify repeated errors and mistakes in their work. Secondly, Iresearched how to address procedural errors and symbolic errors. Through myresearch, I identified several techniques which include: introducing the conceptbefore the procedure, building networks, concrete manipulatives, real-worldapplications, using a variety of formats and representations, reflection and self-assessment, and more time. This research served as guidelines in formattingthe activities that I created to address common mathematical errors.

Formatting Activities that Help Correct and PreventStudents’ Mathematical Errors

Based on my research, it was evident that there are several factors that must beconsidered, such as conceptual learning, representation, and time when help-ing to correct and prevent mathematical errors within a middle school or highschool classroom. As a result, when developing activities that address errorsin a mathematics classroom, it was helpful to consider the following three fac-tors: preconceptions, competence in mathematics, and metacognition. Forexample, students develop preconceptions about mathematical concepts beforethey develop a true conceptual understanding of the concepts. Therefore, thefirst activity when addressing a common mathematical error should engage thestudents’ pre-existing conceptual understanding to help the students recognizeany erroneous preconceptions. Then, students need to show competence in theidentified area of mathematics; therefore, in the initial activities addressing theerror, the students should be asked to relate facts, patterns, and connectionsto the mathematical concepts. In addition, students should be asked to write,illustrate, or state their understanding so that they begin to create their ownnetworks of connections and understanding. These networks will help students

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easily retrieve the information for future use. Lastly, throughout the com-pletion of each activity, the students should reflect upon their understanding.Students must be able to recognize inappropriate connections they have made.In addition, students must be able to relate any new knowledge to their pre-existing knowledge. Also, an overall self-assessment activity that addresses amathematical error would be beneficial. In general, students need to be ableto evaluate their own understanding.

Reflection

While completing my thesis, I developed a stronger conceptual understandingof various mathematical concepts such as fractions, decimals, inequalities, andpolynomials. I researched different representations to include on my activitysheets which addressed various mathematical errors. While exploring these dif-ferent representations, I expanded my own knowledge base. In addition, whiledeveloping my activity sheets, I learned how to put together different represen-tations to illustrate mathematical concepts in several different formats. In myresearch, I found that errors most often occur because not enough examplesare considered. I am hoping that my knowledge of different mathematical rep-resentations will help me include numerous examples in future lessons to helpprevent students from making mathematical errors in my classroom. Also,while working on my thesis, I came to realize how important it is to addresserrors within the classroom. Many of the errors that I addressed in my thesiswere related to basic mathematical concepts that are the foundations on whichstudents are expected to build their knowledge base as they progress throughschool. If these errors are not addressed, students will be trying to build theirknowledge base of mathematics atop misunderstood concepts, which is notlikely to prove successful.

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